A semi-smooth Newton method for magnetic field problems with hysteresis

TL;DR

This paper develops a semi-smooth Newton method for solving nonlinear magnetic field problems involving hysteresis, ensuring well-posedness and demonstrating convergence through theoretical analysis and numerical tests.

Contribution

It introduces a semi-smooth Newton method tailored for hysteresis-including magnetic problems, with proven convergence and applicability to complex models.

Findings

Hysteresis operators are strongly monotone and Lipschitz continuous.

The semi-smooth Newton method converges globally linearly and locally superlinearly.

Numerical tests confirm the theoretical convergence results.

Abstract

Ferromagnetic materials exhibit anisotropy, saturation, and hysteresis. We here study the incorporation of an incremental vector hysteresis model representing such complex behavior into nonlinear magnetic field problems both, from a theoretical and a numerical point of view. We show that the hysteresis operators, relating magnetic fields and fluxes at every material point, are strongly monotone and Lipschitz continuous. This allows to ensure well-posedness of the corresponding magnetic field problems and appropriate finite element discretizations thereof. We further show that the hysteresis operators are semi-smooth, derive a candidate for their generalized Jacobians, and establish global linear and local superlinear convergence of a the semi-smooth Newton method with line search applied to the iterative solution of the discretized nonlinear field problems. The results are proven in detail for a hysteresis model involving a single pinning force and the scalar potential formulation of magnetostatics. The extension to multiple pinning forces and the vector potential formulation is possible and briefly outlined. The theoretical results are further illustrated by numerical tests.